Step 1: A diphtheria carrier harbours Corynebacterium diphtheriae without active disease. The aim is to eradicate the organism from the throat, so an antibiotic is required - not antitoxin (antitoxin neutralises circulating toxin in active disease, not the carrier state).
Step 2: Carriers and close contacts are treated with either benzathine penicillin G (600,000 units if younger than 6 years, 1,200,000 units if 6 years or older) OR oral erythromycin (40 mg/kg/day for children, 1 g/day for adults) for 7 to 10 days.
Step 3: Because both penicillin and erythromycin are accepted regimens, the single best option here is penicillin or erythromycin.
Step 4: Antitoxin (option b) is wrong for a carrier, and ciprofloxacin (option d) is not the standard agent, so the answer is option c.